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In this paper, we determine the $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness of the bilinear Hilbert transform $H_{gamma}(f,g)$ along a convex curve $gamma$ $$H_{gamma}(f,g)(x):=mathrm{p.,v.}int_{-infty}^{infty}f(x-t)g(x-gamma(t)) ,frac{textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $frac{1}{p}+frac{1}{q}=frac{1}{r}$, and $r>frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_{gamma}(f,g)$ along a convex curve $gamma$ $$M_{gamma}(f,g)(x):=sup_{varepsilon>0}frac{1}{2varepsilon}int_{-varepsilon}^{varepsilon}|f(x-t)g(x-gamma(t))| ,textrm{d}t.$$
We establish an L^2 times L^2 to L^1 estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multi-linear oscillatory integrals.
We study the bilinear Hilbert transform and bilinear maximal functions associated to polynomial curves and obtain uniform $L^r$ estimates for $r>frac{d-1}{d}$ and this index is sharp up to the end point.
The Hilbert transforms associated with monomial curves have a natural non-isotropic structure. We study the commutator of such Hilbert transforms and a symbol $b$ and prove the upper bound of this commutator when $b$ is in the corresponding non-isotr
In this paper, for general plane curves $gamma$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(mathbb{R}^2)$-boundedness of the Hilbert transforms $H^infty_{U,gamma}$ along the variable plane curves $(
In this paper, for $1<p<infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^{gamma}$ along a variable plane curve $(t,u(x_1, x_2)gamma(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $gamma$ is a general curve s